Suppose \(O\)  and \(Y\)  are surfaces, with \(Y\)  either a sphere or a torus. In this case, \(\mathcal {C}_{d,r}(O,Y)\)  has the cancellation, inclusion and intersection properties; see Section  and Appendix . Moreover, mapping class groups of compact surfaces are of type \(F_\infty \)  [1]}. Finally, building up on a result of Hatcher–Vogtmann [2]} we will be able to prove in Section  that \(\mathcal {P}(M,A)\)  has the desired connectivity properties. Thus, we have:
